Aharonov-Bohm Interference and Beating in Single-Walled Carbon-Nanotube Interferometers

Relatively low magnetic fields applied parallel to the axis of a chiral single-walled carbon nanotube are found causing large modulations to the $p$ channel or valence band conductance of the nanotube in the Fabry-Perot interference regime. Beating in the Aharonov-Bohm type of interference between two field-induced nondegenerate subbands of spiraling electrons is responsible for the observed modulation with a pseudoperiod much smaller than that needed to reach the flux quantum ${\Phi }_0=h/e$ through the nanotube cross section. We show that single-walled nanotubes represent the smallest cylinders exhibiting the Aharonov-Bohm effect with rich interference and beating phenomena arising from well-defined molecular orbitals reflective of the nanotube chirality.

Figure 1

A suspended chiral-nanotube quantum wire. (a) Schematic device structure. Nanotubes were synthesized across Pt electrodes over trenches at $800–820°\mathrm{C}$ to produce SWNTs with diameters $d<2\mathrm{n}\mathrm{m}$. (b) Electron micrographs of the device layout (left image) and actual suspended nanotube (right image, $L\sim 815\mathrm{n}\mathrm{m}$) used for this work. (c) A resonance micro-raman spectrum (Renishaw, laser $\lambda =785\mathrm{n}\mathrm{m}$, spot size $1\mu m$ scanned over the trench) showing the radial breathing mode (RBM) of a $d\sim 1.5\mathrm{n}\mathrm{m}$ SWNT with possible chirality assignments of $(11,11)$, $(14,8)$, and $(15,6)$ based on the RBM shift at $\varpi =163\pm 4\mathrm{c}{\mathrm{m}}^{-1}$. (d) $G\mathrm{\text{−}}{V}_g$ characteristics recorded at $T=300\mathrm{m}\mathrm{K}$ in a ${}^3\mathrm{H}\mathrm{e}$ cryostat under bias $V=1\mathrm{m}\mathrm{V}$. Left inset: $G$ (represented by color, dark blue: $\sim 2e^2/h$, bright white: $\sim 2.4e^2/h$) vs bias $V$ and $V_g$ for the $p$ channel showing Fabry-Perot interference pattern. Right inset: $G$ vs $V$ and $V_g$ for the $n$ channel displaying Coulomb blockade diamonds.

Figure 2

Experimental data of a nanotube interferometer in magnetic fields. (a) $G\mathrm{\text{−}}{V}_g$ characteristics for the suspended SWNT in magnetic fields (angle to tube axis $\sim 9°$) from 0 to 8 T recorded at $T=300\mathrm{m}\mathrm{K}$ under $V=1\mathrm{m}\mathrm{V}$. The conductance peaks monotonically decrease from 0 to 8 T. (b) A zoom-in view of (a). From top to bottom curves, $G\mathrm{\text{−}}{V}_g$ characteristics in field $B=0$ to 8 T, in 2 T steps. Notice slight shifts in the peak positions to the left at higher fields. (c) A plot of $G$ (represented by color) vs $V_g$ and magnetic field $B$ ($-8\mathrm{T}$ to 8 T) based on 160 $G\mathrm{\text{−}}{V}_g$ curves from $B=-8\mathrm{T}$ to 8 T in 0.1 T steps. Color scale bar unit: $e^2/h$. The slight shifts of conductance peaks positions in (b) are reflected in the slight arching of the interference stripes as highlighted by the dashed line.

Figure 3

The AB effect in a multimode nanotube interferometer. (a) The first Brillouin zone of a small-gap chiral SWNT (in zero magnetic field). Only two parallel lines (dashed) of the allowed states closest to the zero band-gap points (the two solid circles near the inequivalent ${\mathbf{K}}_{\mathbf{1}}$ and ${\mathbf{K}}_{\mathbf{2}}$ corner points, respectively) are shown. Perturbations cause the zero band-gap states deviating from ${\mathbf{K}}_{\mathbf{1}}$ and ${\mathbf{K}}_{\mathbf{2}}$ by $\pm \Delta k_{\perp }{}^0$ due to symmetry and the opening of a small gap along the two dashed lines (${\mathbf{K}}_{\mathbf{1}}$ and ${\mathbf{K}}_{\mathbf{2}}$ subbands). (b) Dispersion $\epsilon (k)$ relations for the two degenerate subbands near ${\mathbf{K}}_{\mathbf{1}}$ (blue curve) and ${\mathbf{K}}_{\mathbf{2}}$ (red curve), respectively. $k_{||}=0$ is defined for the states where the conduction and valence band are the closest. (c) Two degenerate modes of spiraling electrons in zero-field. (d) In a magnetic field parallel to the tube axis, the electronic states (dashed lines) shift by $\Delta k_{AB}$ due to the AB effect, leading to lifting of the degeneracy between the two subbands. (e) Dispersion curves for the two nondegenerate subbands in a magnetic field of $B=8\mathrm{T}$. (f) Two nondegenerate modes of spiraling electrons in a magnetic field.

Figure 4

Simulation for various chirality nanotubes. (a), (b), and (c) are simulation results for $(15,6)$, $(14,8)$, and $(11,11)$ SWNT, respectively. Left panel: Calculated $G\mathrm{\text{−}}{V}_g$ curves in fields of $B=0$ to 8 T from top to bottom in 2 T steps. Middle panel: Calculated 2D plot of $G$ vs $V_g$ and $B$ in a small field range $-8\mathrm{T}$ to 8 T. Right panel: Calculated 2D plot of $G$ vs $V_g$ and $B$ over a wider field range of $-30\mathrm{T}$ to 30 T showing that shifting or arching of the conductance peaks is universal for the three different chirality nanotubes. Our simulations here used precise $\epsilon (k)$ dispersions such as Eqs. (2, 3) instead of linear approximations. Note that we have also considered and excluded the possibility of a $(16,4)$ tube in the experimental sample. The $(16,4)$ tube has a chiral angle of $19°$, larger than the three tubes considered in this figure, and exhibit more rapid beating period than the experimental data.